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An application of the inequality for modified Poisson kernel

Journal of Inequalities and Applications20162016:24

  • Received: 14 September 2015
  • Accepted: 4 January 2016
  • Published:


As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.


  • growth property
  • Dirichlet problem
  • modified Poisson kernel

1 Introduction and results

Let \(\mathbf{R}^{n}\) (\(n\geq2\)) be the n-dimensional Euclidean space. The boundary and the closure of a set E in \(\mathbf{R}^{n}\) are denoted by ∂E and , respectively. The Euclidean distance of two points P and Q in \(\mathbf{R}^{n}\) is denoted by \(\vert P-Q\vert \). Especially, \(\vert P\vert \) denotes the distance of two points P and O in \(\mathbf{R}^{n}\), where O is the origin in \(\mathbf{R}^{n}\).

We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).

Let \(B(P,r)\) denote the open ball with center at P and radius r (>0) in \(\mathbf{R}^{n}\). The unit sphere and the upper half unit sphere in \(\mathbf{R}^{n} \) are denoted by \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\), respectively. The surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \(\mathbf{S}^{n-1}\) is denoted \(w_{n}\). Let \(\Omega\subset\mathbf{S}^{n-1}\), a point \((1,\Theta)\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) are denoted Θ and Ω, respectively. For two sets \(\Lambda\subset\mathbf{R}_{+}\) and \(\Omega\subset\mathbf{S}^{n-1}\), we denote \(\Lambda\times\Omega=\{ (r,\Theta)\in\mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Omega\}\), where \(\mathbf{R}_{+}\) is the set of all positive real numbers.

For the set \(\Omega\subset\mathbf{S}^{n-1}\), we denote the set \(\mathbf{ R}_{+}\times\Omega\) in \(\mathbf{R}^{n}\) by \(C_{n}(\Omega)\), which is called a cone. For the set \(I\subset \mathbf{R}\), the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) are denoted \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\), respectively, where R is the set of all real numbers. Especially, the set \(S_{n}(\Omega; \mathbf{R}_{+})\) is denoted \(S_{n}(\Omega)\).

Given a continuous function f on \(S_{n}(\Omega)\), we say that h is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with f, if h is a harmonic function in \(C_{n}(\Omega)\) and
$$\lim_{P\rightarrow Q\in S_{n}(\Omega), P\in C_{n}(\Omega)}h(P)=f(Q). $$
Let \(\Omega\subset\mathbf{S}^{n-1}\) and \(\Delta^{*}\) be a Laplace-Beltrami on the unit sphere. Consider the Dirichlet problem (see, e.g. [2], p.41)
$$\begin{aligned}& \Delta^{*}\varphi(\Theta)+\lambda\varphi(\Theta)=0 \quad \text{in } \Omega, \\& \varphi(\Theta)=0 \quad \text{in } \partial{\Omega}. \end{aligned}$$
We denote the non-decreasing sequence of positive eigenvalues of it, repeating accordingly to their multiplicities, and the corresponding eigenfunctions are denoted, respectively, by \(\{\lambda_{i}\}_{i=1}^{\infty}\) and \(\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}\). Especially, we denote the least positive eigenvalue of it \(\lambda_{1}\) and the normalized positive eigenfunction to \(\lambda_{1}\) \(\varphi_{1}(\Theta)\). In the sequel, for the sake of brevity, we shall write λ and φ instead of \(\lambda_{1}\) and \(\varphi_{1}\), respectively.
The set of sequential eigenfunctions corresponding to the same value of \(\{\lambda_{i}\}_{i=1}^{\infty}\) in the sequence \(\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}\) makes an orthonormal basis for the eigenspace of the eigenvalue \(\lambda_{i}\). Hence for each \(\Omega\subset S^{n-1}\) there is a sequence \(\{k_{j}\}\) of positive integers such that \(k_{1}=1\), \(\lambda_{k_{j}}<\lambda_{k_{j+1}}\), \(\lambda_{k_{j}}=\lambda_{k_{j}+1}=\lambda_{k_{j}+2}=\cdots=\lambda_{k_{j+1}-1}\) and \(\{\varphi_{k_{j}},\varphi_{k_{j}+1},\ldots,\varphi_{k_{j+1}-1}\}\) is an orthonormal basis for the eigenspace of the eigenvalue \(\{\lambda_{k_{j}}\}_{j=1}^{\infty}\). By \(I_{\Omega}(k_{m})\) we denote the set of all positive integers less than \(\{k_{m}\}_{m=1}^{\infty}\). In spite of the fact
$$I_{\Omega}(k_{1})=\varnothing, $$
the summation over \(I_{\Omega}(k_{1})\) of a function \(S(k)\) of a variable k will be used by promising
$$\sum_{k\in I_{\Omega}(k_{1})}S(k)=0. $$
If we denote the solutions of the equation
$$t^{2}+(n-2)t-\lambda_{i}=0\quad (i=1,2,3,\ldots) $$
by \(\aleph_{i}^{+}\) and \(\aleph_{i}^{-}\), then the functions
$$r^{\aleph_{i}^{\pm}}\varphi_{i}(\Theta) \quad (i=1,2,3,\ldots) $$
are harmonic functions in \(C_{n}(\Omega)\) and vanish on \(S_{n}(\Omega)\).
Let \(G_{\Omega}(P,Q)\) be the Green function of \(C_{n}(\Omega)\) for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in C_{n}(\Omega)\). Then the Poisson kernel in \(C_{n}(\Omega)\) can be defined by
$$PI_{\Omega}(P,Q)=\frac{1}{c_{n}}\frac{\partial}{\partial n_{Q}}G_{\Omega}(P,Q), $$
where \(P\in C_{n}(\Omega)\), \(Q\in S_{n}(\Omega)\), \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(C_{n}(\Omega)\) and
$$c_{n}=\textstyle\begin{cases} 2\pi & \mbox{if } n=2, \\ (n-2)w_{n} & \mbox{if } n\geq3. \end{cases} $$
Let \(F(\Theta)\) be a function defined in Ω. We denote \(N_{i}(F)\) by
$$\int_{\Omega}F(\Theta)\varphi_{i}(\Theta)\,d\Omega, $$
when it exists.
For any two points \(P=(r,\Theta) \) and \(Q=(t,\Phi)\) in \(C_{n}(\Omega)\) and \(S_{n}(\Omega)\), respectively, we define
$$\widetilde{K}_{\Omega}^{m}(P,Q)=\textstyle\begin{cases} 0 & \mbox{if } 0< t< 1, \\ K_{\Omega}^{m}(P,Q) & \mbox{if } 1\leq t< \infty, \end{cases} $$
where m is a non-negative integer and
$$K_{\Omega}^{m}(P,Q)=\sum_{i\in I_{k_{m+1}}}2^{\aleph _{i}^{+}+n-1}N_{i} \bigl(PI_{\Omega}\bigl((1,\Theta),(2,\Phi)\bigr)\bigr)r^{\aleph _{i}^{+}}t^{-\aleph_{i}^{+}-n+1} \varphi_{i}(\Theta). $$
To obtain the solution of the Dirichlet problem in a cone, as in [1, 3, 4], we use the modified Poisson kernel defined by
$$PI_{\Omega}^{m}(P,Q)=PI_{\Omega}(P,Q)- \widetilde{K}_{\Omega}^{m}(P,Q), $$
where \(P\in C_{n}(\Omega)\) and \(Q\in S_{n}(\Omega)\), which has the following estimates (see [1]):
$$ \bigl\vert PI_{\Omega}(P,Q)-K_{\Omega}^{m}(P,Q) \bigr\vert \leq M(2r)^{\aleph_{k_{m+1}}^{+}}t^{-\aleph_{k_{m+1}}^{+}-n+1} $$
for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in S_{n}(\Omega)\) satisfying \(0<\frac{r}{t}<\frac{1}{2}\), where M is a constant independent of P, Q, and m. For the construction and applications of a modified Green function in a half space, we refer the reader to the paper by Qiao (see [5]).
$$U_{\Omega}^{m}[f](P)= \int_{S_{n}(\Omega)}PI_{\Omega}^{m}(P,Q)f(Q)\,d \sigma_{Q}, $$
where \(f(Q)\) is a continuous function on \(\partial C_{n}(\Omega)\) and \(d\sigma_{Q}\) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(\partial{C_{n}(\Omega)}\).

Recently, Qiao and Deng (cf. [1]) gave the solution of the Dirichlet problem in a cone. Applications of modified Poisson kernel with respect to the Schrödinger operator, we refer the reader to the papers by Huang and Ychussie (see [6]) and Li and Ychussie (see [7]).

Theorem A

If \(\Omega+\aleph^{+}-1>0\), \(\Omega-n+1\leq\aleph_{k_{m+1}}^{+}<\Omega-n+2\) and \(f(Q)\) (\(Q=(t,\Phi )\)) is a continuous function on \(\partial{C_{n}(\Omega)}\) satisfying
$$ \int_{S_{n}(\Omega)}\frac{\vert f(Q)\vert }{1+t^{\Omega}}\,d\sigma_{Q}< \infty, $$
then the function \(U_{\Omega}^{m}[f](P)\) is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with f and
$$\lim_{r\rightarrow\infty, P=(r,\Theta)\in C_{n}(\Omega)}r^{n-\Omega-1}\varphi^{n-1}(\Theta) U_{\Omega}^{m}[f](P)=0. $$

Furthermore, Qiao and Deng (cf. [4]) supplemented the above result and proved the following.

Theorem B

Let \(0< p<\infty\), \(\gamma>(-\aleph^{+}-n+2)p+n-1\) and
$$\frac{\gamma-n+1}{p}< \aleph_{k_{m+1}}^{+}< \frac{\gamma-n+1}{p}+1. $$
If \(f(Q)\) (\(Q=(t,\Phi)\)) is a continuous function on \(S_{n}(\Omega)\) satisfying
$$ \int_{S_{n}(\Omega)}\frac {\vert f(Q)\vert ^{p}}{1+t^{\gamma}}\,d\sigma_{Q}< \infty, $$
then the function \(U_{\Omega}^{m}[f](P)\) satisfies
$$\lim_{r\rightarrow\infty, P=(r,\Theta)\in C_{n}(\Omega)}r^{\frac{n-\gamma-1}{p}}\varphi^{n-1}(\Theta) U_{\Omega}^{m}[f](P)=0. $$
It is natural to ask if the continuous function u satisfying (2) and (3) can be replaced by arbitrary continuous function? In this paper, we shall give an affirmative answer to this question. To do this, we first construct a modified Poisson kernel. Let \(\phi(l)\) be a positive function of \(l\geq1\) satisfying
$$2^{\aleph^{+}}\phi(1)=1. $$
Denote the set
$$\bigl\{ l\geq1;-\aleph_{k_{i}}^{+}\log2=\log \bigl(l^{n-1}\phi(l)\bigr)\bigr\} $$
by \(\pi_{\Omega}(\phi,i)\). Then \(1\in\pi_{\Omega}(\phi,i)\). When there is an integer N such that \(\pi_{\Omega}(\phi,N)\neq\Phi\) and \(\pi_{\Omega}(\phi,N+1)= \Phi \), denote
$$J_{\Omega}(\phi)=\{i;1\leq i\leq N\} $$
of integers. Otherwise, denote the set of all positive integers by \(J_{\Omega}(\phi)\). Let \(l(i)=l_{\Omega}(\phi,i+1)\) be the minimum elements l in \(\pi _{\Omega}(\phi,i)\) for each \(i\in J_{\Omega}(\phi)\). In the former case, we put \(l{(N+1)}=\infty\). Then \(l(1)=1\). The kernel function \(\widetilde{K}_{\Omega}^{\phi}(P,Q)\) is defined by
$$\widetilde{K}_{\Omega}^{\phi}(P,Q)=\textstyle\begin{cases} 0 & \mbox{if } 0< t< 1, \\ K_{\Omega}^{i}(P,Q) & \mbox{if } l(i)\leq t< l(i+2) \text{ and } i\in J_{\Omega}(\phi), \end{cases} $$
where \(P\in C_{n}(\Omega)\) and \(Q=(t,\Phi)\in S_{n}(\Omega)\).
The generalized Poisson kernel \(P_{\Omega}^{\phi}(P,Q)\) is defined by
$$PI_{\Omega}^{\phi}(P,Q)=PI_{\Omega}(P,Q)- \widetilde{K}_{\Omega}^{\phi}(P,Q), $$
where \(P\in C_{n}(\Omega)\) and \(Q\in S_{n}(\Omega)\).

As an application of the inequality (1) and the generalized Poisson kernel \(PI_{\Omega}^{\phi}(P,Q)\), we have the following.


Let \(g(Q)\) be a continuous function on \(S_{n}(\Omega)\). Then there is a positive continuous function \(\phi_{g}(l)\) of \(l\geq0\) depending on g such that
$$H_{\Omega}^{\phi_{g}}(P)= \int_{S_{n}(\Omega)}PI_{\Omega}^{\phi _{g}}(P,Q)g(Q)\,d \sigma_{Q} $$
is a solution of the Dirichlet problem in \(C_{n}(\Omega)\) with g.

2 Lemmas

Lemma 1

Let \(\phi(l)\) be a positive continuous function of \(l\geq1\) satisfying
$$\phi(1)=2^{-\aleph^{+}}. $$
$$\bigl\vert PI_{\Omega}(P,Q)-\widetilde{K}_{\Omega}^{\phi}(P,Q) \bigr\vert \leq M \phi(l) $$
for any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any \(Q=(t,\Phi)\in S_{n}(\Omega)\) satisfying
$$ t>\max\{1,4r\}. $$


We can choose two points \(P=(r,\Theta)\in C_{n}(\Omega)\) and \(Q=(t,\Phi)\in S_{n}(\Omega)\), which satisfies (4). Moreover, we also can choose an integer \(i=i(P,Q)\in J_{\Omega}(\phi)\) such that
$$ l(i-1)\leq t< l(i). $$
$$\widetilde{K}_{\Omega}^{\phi}(P,Q)=K_{\Omega}^{i-1}(P,Q). $$
Hence we have from (1), (4), and (5) that
$$\bigl\vert PI_{\Omega}(P,Q)-\widetilde{K}_{\Omega}^{\phi}(P,Q) \bigr\vert \leq M 2^{-\aleph _{k_{i}}^{+}} \leq M \phi(l), $$
which is the conclusion. □

Lemma 2

(See [4])

Let \(g(Q)\) be a continuous function on \(\partial{C_{n}(\Omega)}\) and \(V(P,Q)\) be a locally integrable function on \(\partial{C_{n}(\Omega)}\) for any fixed \(P\in C_{n}(\Omega)\), where \(Q\in \partial{C_{n}(\Omega)}\). Define
$$W(P,Q)=PI_{\Omega}(P,Q)-V(P,Q) $$
for any \(P\in C_{n}(\Omega)\) and any \(Q\in\partial{C_{n}(\Omega)}\).

Suppose that the following two conditions are satisfied:

(I) For any \(Q'\in\partial{C_{n}(\Omega)}\) and any \(\epsilon>0\), there exists a neighborhood \(B(Q')\) of \(Q'\) such that
$$ \int_{S_{n}(\Omega;[R,\infty))}\bigl\vert W(P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \,d\sigma_{Q}< \epsilon $$
for any \(P=(r,\Theta)\in C_{n}(\Omega)\cap B(Q')\), where R is a positive real number.
(II) For any \(Q'\in\partial{C_{n}(\Omega)}\), we have
$$ \limsup_{P\rightarrow Q', P\in C_{n}(\Omega)} \int_{S_{n}(\Omega ;(0,R))}\bigl\vert V(P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \,d\sigma_{Q}=0 $$
for any positive real number R.
$$\limsup_{P\rightarrow Q', P\in C_{n}(\Omega)} \int_{S_{n}(\Omega )}W(P,Q)u\bigl(Q'\bigr)\,d\sigma_{Q}\leq u(Q) $$
for any \(Q'\in\partial{C_{n}(\Omega)}\).

3 Proof of Theorem

Take a positive continuous function \(\phi(l)\) (\(l\geq1\)) such that
$$ \phi(1)2^{\aleph^{+}}=1 $$
$$\phi(l) \int_{\partial\Omega}\bigl\vert g(l,\Phi)\bigr\vert \,d \sigma_{\Phi}\leq\frac{L}{l^{n}} $$
for \(l>1\), where
$$L= 2^{-\aleph^{+}} \int_{\partial\Omega}\bigl\vert g(1,\Phi)\bigr\vert \,d \sigma_{\Phi}. $$
For any fixed \(P=(r,\Theta)\in C_{n}(\Omega)\), we can choose a number R satisfying \(R>\max\{1,4r\}\). Then we see from Lemma 1 that
$$\begin{aligned} & \int_{S_{n}(\Omega;(R,\infty))}\bigl\vert PI_{\Omega}^{\phi_{g}}(P,Q) \bigr\vert \bigl\vert g(Q)\bigr\vert \,d\sigma _{Q} \\ &\quad \leq M \int_{R}^{\infty} \biggl( \int_{\partial\Omega}\bigl\vert g(1,\Phi)\bigr\vert \,d \sigma_{\Phi} \biggr)\phi(l)l^{n-2}\,dl \\ &\quad \leq\quad ML \int_{R}^{\infty} l^{-2}\,dl \\ &\quad < \infty. \end{aligned}$$
Obviously, we have
$$\int_{S_{n}(\Omega;(0,R))}\bigl\vert PI_{\Omega}^{\phi_{g}}(P,Q) \bigr\vert \bigl\vert g(Q)\bigr\vert \,d\sigma _{Q}< \infty, $$
which gives
$$\int_{S_{n}(\Omega)}\bigl\vert PI_{\Omega}^{\phi_{g}}(P,Q) \bigr\vert \bigl\vert g(Q)\bigr\vert \,d\sigma_{Q}< \infty. $$

To see that \(H_{\Omega}^{\phi_{g}}(P)\) is a harmonic function in \(C_{n}(\Omega)\), we remark that \(H_{\Omega}^{\phi_{g}}(P)\) satisfies the locally mean-valued property by Fubini’s theorem.

Finally we shall show that
$$\lim_{P\in C_{n}(\Omega),P\rightarrow Q'}H_{\Omega}^{\phi_{g}}(P)=g \bigl(Q'\bigr) $$
for any \(Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}\). Set
$$V(P,Q)=\widetilde{K}_{\Omega}^{\phi_{g}}(P,Q) $$
in Lemma 2, which is locally integrable on \(S_{n}(\Omega)\) for any fixed \(P\in C_{n}(\Omega)\). Then we apply Lemma 2 to \(g(Q)\) and \(-g(Q)\).

For any \(\epsilon>0\) and a positive number δ, by (9) we can choose a number R (\(>\max\{1,2(t'+\delta)\}\)) such that (6) holds, where \(P\in C_{n}(\Omega)\cap B(Q',\delta)\).

$$\lim_{\Theta\rightarrow \Phi'}\varphi_{i}(\Theta)=0\quad (i=1,2,3\ldots) $$
as \(P=(r,\Theta)\rightarrow Q'=(t',\Phi')\in S_{n}(\Omega)\), we have
$$\lim_{P\in C_{n}(\Omega),P\rightarrow Q'} \widetilde{K}_{\Omega}^{\phi_{g}}(P,Q)=0, $$
where \(Q\in S_{n}(\Omega)\) and \(Q'\in S_{n}(\Omega)\). Then (7) holds.

Thus we complete the proof of Theorem.



The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions.

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Authors’ Affiliations

School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450002, China
Foundation Department, Wuhai Vocational and Technical College, Wuhai, 016000, China


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